Nader M., Kaltenbacher M., Krommer M., Von Garssen H., Lerch R.

In many engineering applications vibrations are responsible for the generation of acoustic noise. Especially slender or thin-walled structures with a large surface contribute to this unwanted noise radiation. In the present paper the concept of dynamic shape control is applied to compensate emerging disturbing vibrations. The goal of shape control is to eliminate structural deformations by means of a distributed control actuation. Thus, the spatial distribution of the actuation, together with its time evolution, is prescribed, such that the total displacement field vanishes throughout the entire structure. The basic idea is derived for the case of a slender cantilever. Due to the prescribed support motion, flexural vibrations are emerging in addition to the rigid body motion. A continuous distribution for the piezoelectric actuation is derived analytically, which is able to eliminate the flexural vibrations of the beam. For practical feasibility the continuous actuation is approximated by distributed piezoelectric patches. The appropriate locations for the actuators as well as for the sensors are determined by finite element computations. If the time variation of the support motion is known an exact elimination can be achieved by the method of dynamic shape control. If this is not the case, dynamic shape control is still a useful tool, as it allows us to design collocated actuators and sensors, which can be used to control the deviation of the deflection. A feedback controller using collocated actuators and sensors is designed and simulated within the environment of MATLAB/Simulink. The performance of the controller is evaluated by applying our finite element software tool, which allows the full simulation of controlled piezoelectric structures.